Jordan's totient function

In number theory, Jordan's totient function $J_{k}(n)$ of a positive integer n, named after Camille Jordan, is defined to be the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. This is a generalisation of Euler's totient function, which is J₁.

Definition

Jordan's totient function is multiplicative and may be evaluated as

$J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right).\,$

Properties

• $\sum _{d|n}J_{k}(d)=n^{k}\,$.
• The average order of J_k(n) is c n^k for some c.

References

• L. E. Dickson (1919, repr.1971). History of the Theory of Numbers I. Chelsea. ISBN 0-8284-0086-5. 
• M. Ram Murty (2001). Problems in Analytic Number Theory. Springer-Verlag. ISBN 0387951431.